Understanding the approximations of mode-coupling theory for sheared steady states of colloids

The lack of clarity of various mode-coupling theory (MCT) approximations, even in equilibrium, makes it hard to understand the relation between various MCT approaches for sheared steady states as well as their regime of validity. Here we try to understand these approximations indirectly by deriving the MCT equations through two different approaches for a colloidal system under shear, first through a microscopic approach, as suggested by Zaccarelli et al., and second through fluctuating hydrodynamics, where the approximations used in the derivation are quite clear. The qualitative similarity of our theory with a number of existing theories show that linear response theory might play a role in various approximations employed in deriving those theories and one needs to be careful while applying them for systems arbitrarily far away from equilibrium, such as a granular system or when shear is very strong. As a by-product of our calculation, we obtain the extension of the Yvon-Born-Green (YBG) equation for a sheared system and under the assumption of random-phase approximation, the YBG equation yields the distorted structure factor that was earlier obtained through different approaches.